Is Bernoulli’s equation applicable for systems with pumps?

Please have patience with me as I have not looked at this sort of theory in many years (I am also very new to StackExchange).

A sloping swimming pool two-thirds full of sea water provides the supply for a fixed installation protecting a risk situated above the pool.

The pool measures 12m long, 6m wide and 1m at the shallow end sloping to 2.5m at the deep end. A pipe is placed so that its inlet is at the bottom of the deep end of the pool. A pump imparts pressure to the water in the pipe such that a mercury manometer indicates a pressure difference of 681mm Hg between inlet and outlet. The inlet of the pipe is 80mm diameter; the outlet nozzle is 45mm in diameter and is 5m above the inlet. The inlet velocity of the water is 5m/s, the density of sea water is 1050kgm-3 and the density of mercury is 13600kgm-3.

Use Bernoulli’s theorem to calculate the velocity of the water through the outlet nozzle?

By applying the continuity equation the following is found:

$$v_1A_1=v_2A_2$$

$$v_2=15.8m/s$$

However, the solution is given as $$v_2=10m/s$$

If I apply Bernoulli’s equation:

$$\frac{P_2 -P_1}{ρ}+\frac{1}{2}(v^2_2-v^2_1)+g(z_2-z_1)=0$$ $$\frac{-90792.5}{1050}+\frac{1}{2}(v^2_2-25)+9.81(5)=0$$

$$v_2=10m/s$$

This is the answer that the solution gives but continuity is not met as shown above. Is this question ill-posed?

I am very confused why continuity is not met when Bernoulli’s equation is used and why the two methods produce different results. A similar question was asked here but this was for a system without a pump and I am not sure if it is applicable.

My question is: Can Bernoulli’s equation be used when the system has a pump or is it no longer valid due to energy not being conserved?

• model the pump as a pressure difference. Jul 27 '21 at 14:50
• The pipe is tilted upward with the outlet located 5m above the inlet, doesn't that makes difference?
– r13
Jul 27 '21 at 18:43

No, there is a problem with the question. If the flow rate is set at the inlet, then that is the flow through the pipe, period. The pump doesn't add mass, just a motive force. Bernouli would apply if you were doing the flow calcs yourself. In this case, they gave you the answer at the beginning. The flow rate through the exit will be equal to the flow rate through the inlet no matter what the pump did or how high the exit is.